Least-squares finite strain hexahedral element/constitutive coupling based on parametrized configurations and the Löwdin frame

P. Areias, C. A. Mota Soares, T. Rabczuk

Research output: Contribution to journalArticlepeer-review

8 Citations (Scopus)


Two novelties are introduced: (i) a finite-strain semi-implicit integration algorithm compatible with current element technologies and (ii) the application to assumed-strain hexahedra. The Löwdin algorithm is adopted to obtain evolving frames applicable to finite strain anisotropy and a weighted least-squares algorithm is used to determine the mixed strain. Löwdin frames are very convenient to model anisotropic materials. Weighted least-squares circumvent the use of internal degrees-of-freedom. Heterogeneity of element technologies introduce apparently incompatible constitutive requirements. Assumed-strain and enhanced strain elements can be either formulated in terms of the deformation gradient or the Green-Lagrange strain, many of the high-performance shell formulations are corotational and constitutive constraints (such as incompressibility, plane stress and zero normal stress in shells) also depend on specific element formulations. We propose a unified integration algorithm compatible with possibly all element technologies. To assess its validity, a least-squares based hexahedral element is implemented and tested in depth. Basic linear problems as well as 5 finite-strain examples are inspected for correctness and competitive accuracy.

Original languageEnglish
Pages (from-to)96-109
Number of pages14
JournalFinite Elements in Analysis and Design
Publication statusPublished - 2016 Jan 1

Bibliographical note

Funding Information:
The authors gratefully acknowledge financing from the " Fundação para a Ciência e a Tecnologia " under the Project PTDC/EME-PME/108751 and the Program COMPETE FCOMP-01-0124-FEDER-010267.

Publisher Copyright:
© 2015 Elsevier B.V. All rights reserved.


  • Assumed-strain hexahedron
  • Constitutiveintegration
  • Finite strains
  • Löwdin frame

ASJC Scopus subject areas

  • Analysis
  • General Engineering
  • Computer Graphics and Computer-Aided Design
  • Applied Mathematics


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